Question

write an equation that can be used to solve for x. solve for x. if necessary, round your answer to the nearest tenth. write an equation that can be used to solve for y.

Explanation:

Step1: Use Pythagorean theorem for lower - right triangle

The lower - right triangle has sides 6 and 8. By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 6\), \(b = 8\), and \(c\) is the hypotenuse of this small triangle. The length of the hypotenuse of the small triangle is \(\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\).

Step2: Use Pythagorean theorem for large right - triangle

The large right - triangle has legs 10 and \(x\) and hypotenuse \(y\).

Equation for \(x\):

The base of the large right - triangle can be found using the fact that the horizontal part of the figure is composed of the base of the small right - triangle and \(x\). Since the base of the small right - triangle is 8, and considering the right - triangle with height 10 and hypotenuse \(y\), by the Pythagorean theorem, the equation to solve for \(x\) is \(x^{2}+10^{2}=y^{2}\). Also, if we consider the overall figure, we can note that the horizontal length from the left - most point to the right - most point of the non - overlapping part of the triangles. The equation for \(x\) using the Pythagorean theorem in the large right - triangle is \(x^{2}+10^{2}=y^{2}\), or we can also consider the relationship between the sides of the combined figure. If we assume the figure is composed of right - triangles, and we know the vertical side of the large right - triangle is 10. Let's assume the length of the base of the large right - triangle is \(x\). The equation for \(x\) based on the Pythagorean theorem is \(x^{2}+10^{2}=y^{2}\). Another way, if we consider the fact that the horizontal length of the non - overlapping part of the triangles, we can say that if we know the length of the hypotenuse of the small triangle is 10 (from step 1), and we consider the large right - triangle with height 10, we can write the equation \(x^{2}+10^{2}=y^{2}\). To solve for \(x\) in terms of known values, we first note that the length of the hypotenuse of the small triangle formed by sides 6 and 8 is 10. Then, in the large right - triangle with height 10, we have \(x^{2}+10^{2}=y^{2}\). If we assume we want to find \(x\) in terms of the given sides, and we know the vertical side of the large right - triangle is 10, and we consider the right - triangle relationship, we get \(x^{2}+10^{2}=y^{2}\), or \(x=\sqrt{y^{2}-100}\). But if we consider the figure in terms of the Pythagorean theorem directly for the large right - triangle, the equation to solve for \(x\) is \(x^{2}+10^{2}=y^{2}\).

Equation for \(y\):

Using the Pythagorean theorem in the large right - triangle with legs 10 and \(x\), the equation is \(y=\sqrt{x^{2}+10^{2}}\) or \(y^{2}=x^{2}+100\).

Answer:

Equation for \(x\): \(x^{2}+10^{2}=y^{2}\)
Equation for \(y\): \(y^{2}=x^{2}+100\)